Dissipative solitons, wave asymmetry and dynamical ratchets

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Abstract

Unidirectional solitonic wave-mediated transport is shown to be possible for a class of anharmonic lattice problems where, due to wave asymmetry, the waves can be used as a traveling periodic ratchet. Using a (mesoscopic) probabilistic description we have assessed the role of both viscous friction and temperature in both the direction of transport and its quantitative features. No asymmetry is required on the potential. Furthermore its actual form and even that of the periodic wave, save its asymmetry, play no significant role in the results obtained and hence they exhibit rather universal value.

Section snippets

Introduction. Wave mediated transport and lattice model-dynamics

Surface waves can be broadly classified as either oscillatory or translatory. Oscillatory waves are periodic in character, imparting to the liquid an undulatory motion with both horizontal and vertical components without causing appreciable displacement. Indeed in each period a fluid particle combines its motion round a circle with forward movement through a distance (Stokes drift) varying as the square of the radius of that circle. Stokes drift represents a second-order correction to the paths

Lattice ring, wave propagation and wave asymmetry

Let us consider a 1D lattice composed of units as shown in Fig. 1. As earlier noted we shall restrict attention to the case with periodic b.c. hence to a lattice ring. It has been shown analytically, numerically and experimentally using the electric analog circuit or computer (Fig. 2) [27], [28] that along such lattice ring periodic quasi-cnoidal waves and solitary waves can stably travel. Fig. 3 depicts one such wave for N=6. This is not the only wave possible. Indeed, with N units in the ring

Traveling solitonic ratchet

Let us now consider the units in the original mechanical lattice ring as heavy ions and let us add a light particle in 3D-geometry to rule out unnecessary singularities due to (1D and 2D) geometry. The light particle can be an “electron”, and hence with opposite albeit equal charge to the ions, the electron–lattice interaction can be taken, e.g., asUe(r)=-U01+r2/h2,orUe(r)=-U0e-r2/2h2,which is a Gaussian well. In both cases, U0 is the maximum and h is the minimal distance allowed between the

The role of noise and temperature upon transport

Let us now discuss the role of noise in the system for values around γ2m and in region A. One may expect that the noise strength alters the direction of motion. For low noise level, particles with γ in the region A, move within a potential minimum (Fig. 5b, m1) with velocity vr>0, in opposite direction to the solitonic wave. After some time lapse, they take on a deeper nearby available minimum (Fig. 5b, m2) and travel with negative speed higher in absolute value and hence in the soliton motion

Concluding remarks

Wave-mediated transport is first-order in nonlinear propagating waves. To implement it we have used this fact together with the ratchet character of dissipative solitons traveling along a lattice with anharmonic (exponentially repulsive) interactions. The specific form of the potential is not significant from the physical standpoint and the main results found. What matters is that such (periodic) solitons have asymmetric wave forms while the underlying potential is, generally, symmetric. As in

Acknowledgments

This research has been sponsored by the European Union under Grant SPARK (FP6-004690) and by the Spanish Ministerio de Educación y Ciencia under Grant ESP2004-01511.

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